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To deblur the image. usually we consider the model $$B=A(X)+e,$$ where $X$ is the expected image, $A$ is the convolution/matrix action and $B$ is the blurred image.

I know there are some matrix-methods to deal with the above case, e.g. in the book Deblurring Images: Matrices, Spectra, and Filtering.

Q: Are there some pde model/ functional model to understand the deblurring problem in mathematics. In other words, could anyone give some reference about the analytic methods to deblurr the image.

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I'll assume the image to be recovered is an $m \times n$ array of real numbers. A very common method is to solve a convex optimization problem such as $$ \operatorname{minimize}_{X \in \mathbb R^{m \times n}} \quad \frac12 \| A(X) - B \|^2 + \gamma \| Dx \|_1 $$ where $D$ is a discrete gradient operator (in which case we're using "total variation regularization") or some sparsifying transformation such as a wavelet transformation. The norm $\| \cdot \|_1$ is just the usual $\ell_1$-norm, and $\| \cdot \|$ is the Frobenius norm.

Here is a great tutorial paper on how to solve image processing problems like this one: "An introduction to continuous optimization for imaging" by Chambolle and Pock. It was published in 2016.

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  • $\begingroup$ @DLIN I edited my answer to clarify that. $\endgroup$
    – littleO
    Jan 22, 2017 at 10:53
  • $\begingroup$ How about the blind deblurring model? Could you also give a reference? $\endgroup$
    – DLIN
    Jan 26, 2017 at 2:56

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